Find the value of k, so that the function
\(f(x)=\begin{cases} { kx }^{ 2 },\quad if\quad x\ge 1 \\ 4\quad ,\quad if\quad x<1 \end{cases}\) is continuous at x=1.

4

(a)

For what value of λ is the function defined by
\(f(x)=\begin{cases} \lambda ({ x }^{ 2 }-2x)\quad ,\ if\ x\le 0 \\ 4x+1\quad \quad \ ,\quad if\ x>0 \end{cases} \)  continuous at x = 0?
What about continuity at x = 1?

4

(a)

For what value of k is the function \(f(x)=\begin{cases} \frac { { e }^{ x }+{ e }^{ -x }-2 }{ x^{ 2 } }\ \ ,if\quad x\neq 0 \\ \quad 4k \ \ \ \ \ \ \ \ \ , if\quad x=0 \end{cases}\)is continuous at x = 0?

4

(a)

For what value of k, is the following function continuous at x=0?
\(f(x)=\begin{cases} \frac { 1-cos\quad 4x }{ 8x^{ 2 } } \ , \ \ if\quad x\neq 0 \\ \quad \quad \quad k \ \ \ , \ \ if\quad x=0 \end{cases}\)

4

(a)

Find the derivative of each of the following function w.r.t. x, or find \(\frac { dy }{ dx } \):
\(y=\sqrt { \frac { sec \ \ x-1 }{ sec \ \ \ x+1 } } \)

4

(a)

If log y= tan^-1 x, show that (1+x²) y₂ + (2x-1) y₁=0

4

(a)

If y= 3e^2x + 2e^3x, prove that \(\frac { d^{ 2 }y }{ dx^{ 2 } } -5\frac { dy }{ dx } +6y=0\)

4

(a)

Differentiate the following w.r.t. x or find \(\frac { dy }{ dx } \).
y=\(\sqrt { a+\sqrt { a+x } } \).

4

(a)

If y= (sin^-1 x)² , prove that (1-x²) \(\frac { { d }^{ 2 }y }{ d{ x }^{ 2 } } -x\frac { dy }{ dx } -2=0.\)

4

(a)

Verify Rolle's Theorem for the following functions:
f(x) = |x| in |-1, 1|

4

(a)